The document starts with a prologue discussing the Ontological aspects of the concept of Space to indicate where in the scheme of human endeavour this work fits. The prologue is followed by a prospectus intended to mitigate the strangeness of the concepts by offering a “bird’s eye view” of some concepts and arguments. This is then followed by a note to the reader containing some background information and precepts. The main discourse “LEIBNIZIAN COSMOLOGY and LEIBNIZIAN MATHEMATICS”, which consists of nine sections, then follows. It starts by introducing the core difference between the Leibnizian Cosmology and the Euclidean Cosmology by way of an example showing that the real line can contain only countable many real numbers in the Leibnizian Cosmology. Though the two cosmologies are formally stated only in paragraphs four and five, the main relevant difference is introduced by replacing the Euclidean assumption that points exist and that a line is a string of points with the Leibnizian assumption that lines exist and that points are merely the ends of lines. Then follows a discussion of some relevant properties of perceived space and how they impact on Ontology. This is followed by an analysis of the basic assumptions of the Euclidean approach showing that these assumptions imply that a line must be formed from more than countable many points and that it is possible to perform infinitely many operations to completion. The relevant basic assumptions of the Leibnizian Cosmology, including the Axiom of Parmenides, are then introduced. Thereafter the basics of Leibnizian Mathematics are developed. Firstly, the standard argument for obtaining the gradient of a straight line to a curve is analysed to come to the concept of infinitesimal as it was introduced by Leibniz. The rule of L’Hospital is transcribed into a form in which the concept of limit is not present and so to come to a definition of an infinitesimal number. The number concept is then extended to the Cauchy Numbers. A special case of the Riemann integral is then analysed to motivate the definition of an infinitesimal. The Fundamental Theorem of Calculus is next analysed from the perspective of Leibnizian Mathematics. The Dirac δ-function is presented as an example of a function that can have an infinite Cauchy number as value.